Toeplitz Algebras and Extensions ofIrrational Rotation Algebras

[摘要] For a given irrational number $heta$, we define Toeplitz operators withsymbols in the irrational rotation algebra ${mathcal A}_heta$,and we show that the $C^*$-algebra $mathcal T({mathcalA}_heta)$ generated by these Toeplitz operators is an extensionof ${mathcal A}_heta$ by the algebra of compact operators. Wethen use these extensions to explicitly exhibit generators of thegroup $KK^1({mathcal A}_heta,mathbb C)$. We also prove anindex theorem for $mathcal T({mathcal A}_heta)$ thatgeneralizes the standard index theorem for Toeplitz operators onthe circle.

[学科分类] 数学（综合）

[关键词] Toeplitz operators;irrational rotation algebras;index theory